Rukasha Maths

GCSE maths Beta

Cumulative frequency and box plots · Revision guide

Higher · Statistics · 1 page in the PDF

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Learning objectives

  • Estimate a median and quartiles from cumulative frequency.
  • Compare the centre and spread of distributions using box plots.

Key ideas

Cumulative frequency is a running total; plot it against upper class boundaries.

For grouped graphs, locate quartiles using these cumulative frequency positions.

Q1:N4,median:N2,Q3:3N4Q_1:\frac N4,\qquad\text{median}:\frac N2,\qquad Q_3:\frac{3N}{4}

The interquartile range measures the spread of the middle half.

IQR=Q3−Q1\mathrm{IQR}=Q_3-Q_1

A box plot shows minimum, lower quartile, median, upper quartile and maximum. Compare centre and spread separately.

Worked examples

Example 1

For 80 journeys, readings at cumulative frequencies 20, 40 and 60 are 18, 26 and 34 minutes. Estimate the median and interquartile range.

804=20,802=40,3(80)4=60\frac{80}{4}=20,\qquad\frac{80}{2}=40,\qquad\frac{3(80)}4=60
Q1≈18,median≈26,Q3≈34 minutesQ_1\approx18,\qquad\text{median}\approx26,\qquad Q_3\approx34\ \text{minutes}
IQR≈34−18\mathrm{IQR}\approx34-18
median≈26 minutes,IQR≈16 minutes\text{median}\approx26\ \text{minutes},\qquad\mathrm{IQR}\approx16\ \text{minutes}

Example 2

The lower quartile is sixteen and the upper quartile twenty-nine. Find the interquartile range.

IQR=Q3−Q1=29−16\mathrm{IQR}=Q_3-Q_1=29-16
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Watch out

Read the median at half the total cumulative frequency. Graph readings are estimates; interquartile range is not the full range.